Theorem condan 481

Description: Proof by contradiction.

Hypotheses

Ref	Expression
condan.1	⊢ ((φ ⋀ ¬ ψ) → χ)
condan.2	⊢ ((φ ⋀ ¬ ψ) → ¬ χ)

Assertion

Ref	Expression
condan	⊢ (φ → ψ)

Proof of Theorem condan

Step	Hyp	Ref	Expression
1		condan.1	. . . 4 ⊢ ((φ ⋀ ¬ ψ) → χ)
2	1	ex 371	. . 3 ⊢ (φ → (¬ ψ → χ))
3		condan.2	. . . 4 ⊢ ((φ ⋀ ¬ ψ) → ¬ χ)
4	3	ex 371	. . 3 ⊢ (φ → (¬ ψ → ¬ χ))
5	2, 4	pm2.65d 134	. 2 ⊢ (φ → ¬ ¬ ψ)
6		notnot2 84	. 2 ⊢ (¬ ¬ ψ → ψ)
7	5, 6	syl 10	1 ⊢ (φ → ψ)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 221

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 145  df-an 223

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